Question

\((x-y)^3+(y-z)^3+(z-x)^3=\)

• A. \(3xyz\)
• B. \((x-y)(y-z)(z-x)\)
• C. \(3(x-y)(y-z)(z-x)\)
• D. 0

Hint:

Whenever three expressions add up to zero, check whether the identity \(a^3+b^3+c^3=3abc\) can be applied.

Answer 1

The Correct Option is C

Step 1: Recall a useful identity.
If three numbers add up to zero, that is $a + b + c = 0$, then \[ a^3 + b^3 + c^3 = 3abc \]

Step 2: Choose smart names.
Let \[ a = x - y, \quad b = y - z, \quad c = z - x \]

Step 3: Add them up.
\[ a + b + c = (x-y) + (y-z) + (z-x) \]

Step 4: Simplify the sum.
All the letters cancel out: \[ x - y + y - z + z - x = 0 \] So the condition $a + b + c = 0$ is true.

Step 5: Use the identity.
Since the sum is zero, \[ a^3 + b^3 + c^3 = 3abc \]

Step 6: Substitute back.
\[ (x-y)^3 + (y-z)^3 + (z-x)^3 = 3(x-y)(y-z)(z-x) \] \[ \boxed{3(x-y)(y-z)(z-x)} \]